StePS (Stereographically Projected Cosmological Simulations) compactifies the infinite spatial extent of the Universe into a finite sphere with isotropic boundary conditions to simulate the evolution of the large-scale structure. This eliminates the need for periodic boundary conditions, which are a numerical convenience unsupported by observation and which modifies the law of force on large scales in an unrealistic fashion. StePS uses stereographic projection for space compactification and naive O(N2) force calculation; this arrives at a correlation function of the same quality more quickly than standard (tree or P3M) algorithms with similar spatial and mass resolution. The N2 force calculation is easy to adapt to modern graphics cards, hence StePS can function as a high-speed prediction tool for modern large-scale surveys.